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Operations Management Supplement 6 – Statistical Process Control © 2006 Prentice Hall, Inc. PowerPoint presentation to accompany Heizer/Render Principles of Operations Management, 6e Operations Management, 8e

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Outline Statistical Process Control (SPC) Control Charts for Variables The Central Limit Theorem Setting Mean Chart Limits (x-Charts) Setting Range Chart Limits (R-Charts) Using Mean and Range Charts Control Charts for Attributes Managerial Issues and Control Charts

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Outline – Continued Process Capability Process Capability Ratio (C p ) Process Capability Index (C pk ) Acceptance Sampling Operating Characteristic Curve Average Outgoing Quality

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Learning Objectives When you complete this supplement, you should be able to: Identify or Define: Natural and assignable causes of variation Central limit theorem Attribute and variable inspection Process control x-charts and R-charts

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Learning Objectives When you complete this supplement, you should be able to: Identify or Define: LCL and UCL P-charts and c-charts C p and C pk Acceptance sampling OC curve

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Learning Objectives When you complete this supplement, you should be able to: Describe or Explain: The role of statistical quality control

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Variability is inherent in every process Natural or common causes Special or assignable causes Provides a statistical signal when assignable causes are present Detect and eliminate assignable causes of variation Statistical Process Control (SPC)

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Natural Variations Also called common causes Affect virtually all production processes Expected amount of variation Output measures follow a probability distribution For any distribution there is a measure of central tendency and dispersion If the distribution of outputs falls within acceptable limits, the process is said to be “in control”

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Assignable Variations Also called special causes of variation Generally this is some change in the process Variations that can be traced to a specific reason The objective is to discover when assignable causes are present Eliminate the bad causes

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Samples To measure the process, we take samples and analyze the sample statistics following these steps (a)Samples of the product, say five boxes of cereal taken off the filling machine line, vary from each other in weight Frequency Weight # ## # ## ## # ### #### ######### # Each of these represents one sample of five boxes of cereal Figure S6.1

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Samples To measure the process, we take samples and analyze the sample statistics following these steps (b)After enough samples are taken from a stable process, they form a pattern called a distribution The solid line represents the distribution Frequency Weight Figure S6.1

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Samples To measure the process, we take samples and analyze the sample statistics following these steps (c)There are many types of distributions, including the normal (bell-shaped) distribution, but distributions do differ in terms of central tendency (mean), standard deviation or variance, and shape Weight Central tendency Weight Variation Weight Shape Frequency Figure S6.1

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Samples To measure the process, we take samples and analyze the sample statistics following these steps (d)If only natural causes of variation are present, the output of a process forms a distribution that is stable over time and is predictable Weight Time Frequency Prediction Figure S6.1

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Samples To measure the process, we take samples and analyze the sample statistics following these steps (e)If assignable causes are present, the process output is not stable over time and is not predicable Weight Time Frequency Prediction?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? Figure S6.1

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Control Charts Constructed from historical data, the purpose of control charts is to help distinguish between natural variations and variations due to assignable causes

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Central Limit Theorem Regardless of the distribution of the population, the distribution of sample means drawn from the population will tend to follow a normal curve 1.The mean of the sampling distribution (x) will be the same as the population mean x = n x =x =x =x = 2.The standard deviation of the sampling distribution ( x ) will equal the population standard deviation ( ) divided by the square root of the sample size, n

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Process Control Figure S6.2 Frequency (weight, length, speed, etc.) Size Lower control limit Upper control limit (a) In statistical control and capable of producing within control limits (b) In statistical control but not capable of producing within control limits (c) Out of control

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Population and Sampling Distributions Three population distributions Beta Normal Uniform Distribution of sample means Standard deviation of the sample means = x = n Mean of sample means = x ||||||| -3 x -2 x -1 x x+1 x +2 x +3 x 99.73% of all x fall within ± 3 x 95.45% fall within ± 2 x Figure S6.3

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Sampling Distribution x = (mean) Sampling distribution of means Process distribution of means Figure S6.4

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Steps In Creating Control Charts 1.Take samples from the population and compute the appropriate sample statistic 2.Use the sample statistic to calculate control limits and draw the control chart 3.Plot sample results on the control chart and determine the state of the process (in or out of control) 4.Investigate possible assignable causes and take any indicated actions 5.Continue sampling from the process and reset the control limits when necessary

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Control Charts for Variables For variables that have continuous dimensions Weight, speed, length, strength, etc. x-charts are to control the central tendency of the process R-charts are to control the dispersion of the process These two charts must be used together

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Setting Chart Limits For x-Charts when we know Upper control limit (UCL) = x + z x Lower control limit (LCL) = x - z x wherex=mean of the sample means or a target value set for the process z=number of normal standard deviations x =standard deviation of the sample means = / n =population standard deviation n=sample size

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Setting Control Limits Hour 1 SampleWeight of NumberOat Flakes 117 213 316 418 517 616 715 817 916 Mean16.1 =1 HourMeanHourMean 116.1715.2 216.8816.4 315.5916.3 416.51014.8 516.51114.2 616.41217.3 n = 9 LCL x = x - z x = 16 - 3(1/3) = 15 ozs For 99.73% control limits, z = 3 UCL x = x + z x = 16 + 3(1/3) = 17 ozs

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17 = UCL 15 = LCL 16 = Mean Setting Control Limits Control Chart for sample of 9 boxes Sample number |||||||||||| 123456789101112 Variation due to assignable causes Variation due to natural causes Out of control

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Setting Chart Limits For x-Charts when we don’t know Lower control limit (LCL) = x - A 2 R Upper control limit (UCL) = x + A 2 R whereR=average range of the samples A 2 =control chart factor found in Table S6.1 x=mean of the sample means

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Control Chart Factors Table S6.1 Sample Size Mean Factor Upper Range Lower Range n A 2 D 4 D 3 21.8803.2680 31.0232.5740 4.7292.2820 5.5772.1150 6.4832.0040 7.4191.9240.076 8.3731.8640.136 9.3371.8160.184 10.3081.7770.223 12.2661.7160.284

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Setting Control Limits Process average x = 16.01 ounces Average range R =.25 Sample size n = 5

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Setting Control Limits UCL x = x + A 2 R = 16.01 + (.577)(.25) = 16.01 +.144 = 16.154 ounces Process average x = 16.01 ounces Average range R =.25 Sample size n = 5 From Table S6.1

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Setting Control Limits UCL x = x + A 2 R = 16.01 + (.577)(.25) = 16.01 +.144 = 16.154 ounces LCL x = x - A 2 R = 16.01 -.144 = 15.866 ounces Process average x = 16.01 ounces Average range R =.25 Sample size n = 5 UCL = 16.154 Mean = 16.01 LCL = 15.866

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R – Chart Type of variables control chart Shows sample ranges over time Difference between smallest and largest values in sample Monitors process variability Independent from process mean

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Setting Chart Limits For R-Charts Lower control limit (LCL R ) = D 3 R Upper control limit (UCL R ) = D 4 R where R=average range of the samples D 3 and D 4 =control chart factors from Table S6.1

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Setting Control Limits UCL R = D 4 R = (2.115)(5.3) = 11.2 pounds LCL R = D 3 R = (0)(5.3) = 0 pounds Average range R = 5.3 pounds Sample size n = 5 From Table S6.1 D 4 = 2.115, D 3 = 0 UCL = 11.2 Mean = 5.3 LCL = 0

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Mean and Range Charts (a) These sampling distributions result in the charts below (Sampling mean is shifting upward but range is consistent) R-chart (R-chart does not detect change in mean) UCLLCL Figure S6.5 x-chart (x-chart detects shift in central tendency) UCLLCL

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Mean and Range Charts R-chart (R-chart detects increase in dispersion) UCLLCL Figure S6.5 (b) These sampling distributions result in the charts below (Sampling mean is constant but dispersion is increasing) x-chart (x-chart does not detect the increase in dispersion) UCLLCL

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Automated Control Charts

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Control Charts for Attributes For variables that are categorical Good/bad, yes/no, acceptable/unacceptable Measurement is typically counting defectives Charts may measure Percent defective (p-chart) Number of defects (c-chart)

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